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Question

If α,β are the roots of the equation x22x+4=0, then the value of α6+β6 is 


Solution

Let f(x)=x22x+4=0 has roots α,β, then the equation whose roots are α6,β6 is
f(x1/6)=0x1/32x1/6+4=0x1/6(x1/62)=4
Cubing on both sides,
x1/2(x1/286x1/6(x1/62))=64x1/2(x1/28+24)=64x+16x=64x+64=16x
Squaring on both the sides,
x2+128x+642=256xx2128x+642=0
Therefore, sum of roots is,
α6+β6=128

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